Duke University Mathematics Department
             presents
John J. Gergen Memorial Lecture Series
          May, 5-7, 2008

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Title:  Optimal Transport and Riemannian Geometry : Monge meets Riemann
Three lectures given by Cedric Villani of the ENS-Lyon

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Lecture 1 - The Monge-Kantorovich problem:

The Monge-Kantorovich problem asks about the most economic way to 
transport matter from one prescribed distribution to another one. Born 
in France around the time of the Revolution, this problem has become a 
classic one in probability and economics. At the end of the eighties, 
the independent works of Brenier, Cullen and Mather announced a sharp 
turn of the theory, with renewed interest by the analysts. The speaker 
will present a summary of the modern theory of this problem.

Monday, May 5, 2008 at 4:00 p.m. in Physics 128

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Lecture 2 - Monge, Boltzmann and Ricci:

Starting from works of Otto and Villani, it was understood that Ricci 
curvature bounds are intimately linked with the behavior of Boltzmann's 
entropy functional along geodesics (in the space of probability measures 
on the manifold of interest) induced by the optimal transport problem. 
This observation can be exploited to give a new point of view of Ricci 
curvature, with probabilistic and geometric applications (one is the 
weak stability of Ricci curvature bounds, which was proven independently 
by Lott and Villani, and by Sturm).

Tuesday, May 6, 2008 at 4:00 p.m. in Physics 119

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Lecture 3 - Regularity, curvature and the cut locus:

The regularity of optimal transport in curved geometry is a singularly 
difficult problem because of the sharp interaction between geometry and 
analysis. In connection with this problem a new curvature tensor has 
been introduced by Ma, Trudinger and Wang; it plays a key role in the 
analysis of the smoothness of optimal transport. As shown in a work with 
Loeper, this tensor also has striking implications about the shape of 
the cut locus.
Wednesday, May 7, 2008 at 4:00 p.m. in Physics 119



Tea will be served prior to each lecture at 3:30 p.m. in Physics 101.

There will also be a reception in Physics 101 immediately following the 
Monday lecture.

For additional information please contact the Mathematics Department at 
919-660-2800.

A poster is available at 
http://www.math.duke.edu/seminars/GergenPoster2008Villani.pdf

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